Resonance Engineering of Quality Factor: The Q-Control Method

Introduction

While wafer batch fabrication of sensors and resonators guaranties high-quality control and mass production, the sensor characteristics such as its resonance frequency f₀ and quality factor Q are permanently set and require a complete redesign of the fabrication process to be changed. Since the Q factor determines the measurement bandwidth f₀/2Q, advanced measuring schemes in different environments – such as liquid, ambient, or vacuum – benefit from different values of Q, hence effectively modifying the measurement bandwidth. The quality factor is also a measure of energy dissipation which is directly related to the sensor performance, and consequently improving Q improves the sensitivity.

In practice, there exist many use cases, especially in the field of Scanning Probe Microscopy (SPM), where lowering the Q factor to improve scanning speed or increasing it to improve sensitivity while tracking its resonance frequency is very valuable.

This blog explains how every user can implement Q-Control feedback on any Zurich Instruments lock-in amplifier, at kHz, MHz or even GHz, provided they have the multifrequency and PID options enabled.

Principle of Q-Control

The Quality factor of a resonator is a measure of the energy loss per oscillation cycle. The Q-control method is therefore equivalent to induce or compensate for losses that occur during this periodic resonator motion. This can be achieved by introducing phase delay and variable gain control of the driving signal to the sensor as depicted in the following diagram:

The frequency of the driving signal is unaffected by the resonance but shifted in phase and amplified in magnitude. The role of the Q-controller consists in rectifying this phase and amplitude to the desired level and re-inject it into the drive signal so that the effective resonance response appears as increased (higher Q-factor) or damped (lower Q-factor). What is the mechanism behind this rectifying procedure?

When no loss or gain is involved, a simple phase shifter is enough to change the amplitude response of 2 incoming waves. This is what happens for instance with the linear superposition of 2 sine waves at the same frequency: in-phase (0°) results in doubling the amplitude, while the same waves out-of-phase (180°), cancels the amplitude altogether. In practice, the same situation occurs while also compensating for dissipation in the loop. This is achieved by adding a drive that let the user set the desired resonator response with pre-defined gain factors. Such positive or negative gains will have the effect to modify both amplitude and phase response of the resonance resulting in a new effective bandwidth (Full Width at Half Maximum) and phase slope. The Q-factor can then be computed, for instance using this formula:

If we check how such feedback affects the effective response of the “native” resonance, the results look like Figure 2, in case of Q-factor enhancement:

This shows that the amplitude response is boosted at the same time as the phase slope gets steeper at resonance. In a similar but reverse fashion by introducing losses and therefore diminishing the effective Q factor, the amplitude and phase response looks like this:

Control of Q factor is therefore equivalent to introducing an additional wave in the resonant loop that is adequately tuned in phase and amplitude with respect to the native resonance. Such a corrected wave using a feedback loop can then either boost the signal or reduce it at a given carrier frequency.

LabOne Implementation

Now that we understand how Q control works, let’s implement it within the LabOne User Interface. For this, two demodulators and PID units are required. The reason is that the drive output cannot simply be fed as a PID input without demodulating it. The resulting phase shifter and variable gain controller are replaced by lock-in demodulators with their own phase and PID loops that act as an amplifier with a simple Proportional P-gain. All of this is referenced to the drive excitation of a quartz resonator in our case. Completing the previous diagram with lock-in and feedback loops now looks like this:

The native resonator response is measured with its own amplitude and phase at a set frequency. In terms of lock-in demodulated outputs, both the in-phase and quadrature components, i.e. \(X=R\cos(\theta)\) \(Y=R\sin(\theta)\), respectively, are measured in real-time. Based on the X and Y components, two feedback loops on the drive amplitude can be set and added to the resonant loop with the same reference frequency but shifted in phase by 90°. In the LabOne User Interface this looks like Figure 5:

From the Output Amplitudes in LabOne, the MF-MD Multi-demodulator option allows for the linear superposition of up to 4 sine waves. In our case, the 3 sine waves are added with the same frequency but with different amplitude and phase, resulting in an effective phase shifter and gain controller. The amplitude of sine waves 3 and 4 are the results of the 2 feedback loops on PID 3 and PID 4 sharing the same P-gain value with positive or negative sign to compensate or enhance the closed loop response. Here is the corresponding LabOne screenshot:

While all those parameters can be set manually, it is easier to start with an example setting file for the LabOne interface since several modules and parameters are used together. After loading the setting file in LabOne, one should adjust the parameters such as resonance frequency to adapt to their specific resonator and experiment. This setting file works with the MFLI but the same principle and capabilities is available with all Zurich Instruments Lock-in Amplifiers (use of 2 extra demodulators and 2 PIDs).

Q-control with Phase-Locked Loop (PLL)

While PID 3 and 4 are running for the generation of the corrected output wave at a fixed oscillator frequency, it remains possible to use PID 1 in PLL mode to track the center frequency at resonance. The usual PLL optimization procedure can be followed by recording the new effective resonance and extracting the Q-factor (controlled), resonance frequency and phase at resonance from the Math tool in the Sweeper module. Those measured values can then be fed in the PID Advisor of PID 1 using DUT Model "Resonator Frequency". For the PLL and Q-control to work together, the PLL closed-loop bandwidth must be higher than the Q-control bandwidth. Indeed, the new oscillator frequency from the PLL must be updated before PID 3 and 4 rectify the new amplitude and phase of the output.

Automation with API

For daily operation and automation, it is recommended to use our Application Programming Interface API that is available, together with LabOne, from Zurich Instruments download center. The following API examples are available for Q-control implementation:

• LabVIEW API (ziExample-QControl.vi available in LabOne installer)
• Python API example
• MATLAB API example

As often recommended, it is preferable to check and validate everything in the LabOne GUI first, to make sure that everything behaves as expected, before automating it.

Conclusion

Controlling the Q-factor of a resonator allows for more flexibility in the design of complex experiments, for which the default sensor characteristics can be limiting. Such optimization process is not trivial, and the flexibility offered by all Zurich Instruments lock-in amplifiers – with its multiple phase shifter and feedback loops - allows for fine tuning and monitoring of all relevant parameters.

In this blog post, we have shown how to setup the LabOne user interface using 2 additional demodulators and PID loops to perform Q-Control on a quartz resonator. This approach allows us to better understand how the method works and optimize it. The MFLI setting file and API examples are available here and any Zurich Instruments user can start implementing it on their own devices with the required options.

Acknowledgements

I would like to thank Mehdi Alem for valuable feedback on the content, drawings and testing of the Python API implementation of the Q-Control method.